Mathematical Modeling and Computation Ser.: Rank Deficient and Discrete Ill-Posed Problems : Numerical Aspects of Linear Inversion by Per Christian Hansen (1997, Trade Paperback)

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RANK-DEFICIENT AND DISCRETE ILL-POSED PROBLEMS: NUMERICAL ASPECTS OF LINEAR INVERSION (MONOGRAPHS ON MATHEMATICAL MODELING AND COMPUTATION) By Per Christian Hansen.

About this product

Product Identifiers

PublisherSociety for Industrial AND Applied Mathematics

ISBN-100898714036

ISBN-139780898714036

eBay Product ID (ePID)397434

Product Key Features

Number of Pages264 Pages

LanguageEnglish

Publication NameRank Deficient and Discrete Ill-Posed Problems : Numerical Aspects of Linear Inversion

Publication Year1997

SubjectGeneral, Mathematical Analysis

TypeTextbook

AuthorPer Christian Hansen

Subject AreaMathematics, Computers

SeriesMathematical Modeling and Computation Ser.

FormatTrade Paperback

Dimensions

Item Height0.8 in

Item Weight17.7 Oz

Item Length9 in

Item Width6 in

Additional Product Features

Intended AudienceScholarly & Professional

LCCN97-032066

Dewey Edition21

IllustratedYes

Dewey Decimal512.9/42

Table Of ContentPreface Symbols and Acronyms Chapter 1: Setting the Stage. Problems With Ill-Conditioned Matrices Ill-Posed and Inverse Problems Prelude to Regularization Four Test Problems Chapter 2: Decompositions and Other Tools. The SVD and its Generalizations Rank-Revealing Decompositions Transformation to Standard Form Computation of the SVE Chapter 3: Methods for Rank-Deficient Problems. Numerical Rank Truncated SVD and GSVD Truncated Rank-Revealing Decompositions Truncated Decompositions in Action Chapter 4. Problems with Ill-Determined Rank. Characteristics of Discrete Ill-Posed Problems Filter Factors Working with Seminorms The Resolution Matrix, Bias, and Variance The Discrete Picard Condition L-Curve Analysis Random Test Matrices for Regularization Methods The Analysis Tools in Action Chapter 5: Direct Regularization Methods. Tikhonov Regularization The Regularized General Gauss-Markov Linear Model Truncated SVD and GSVD Again Algorithms Based on Total Least Squares Mollifier Methods Other Direct Methods Characterization of Regularization Methods Direct Regularization Methods in Action Chapter 6: Iterative Regularization Methods. Some Practicalities Classical Stationary Iterative Methods Regularizing CG Iterations Convergence Properties of Regularizing CG Iterations The LSQR Algorithm in Finite Precision Hybrid Methods Iterative Regularization Methods in Action Chapter 7: Parameter-Choice Methods. Pragmatic Parameter Choice The Discrepancy Principle Methods Based on Error Estimation Generalized Cross-Validation The L-Curve Criterion Parameter-Choice Methods in Action Experimental Comparisons of the Methods Chapter 8. Regularization Tools Bibliography Index.

SynopsisHere is an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, seismology, astronomy, and other areas. Rank-deficient problems involve matrices that are exactly or nearly rank deficient. Such problems often arise in connection with noise suppression and other problems where the goal is to suppress unwanted disturbances of given measurements. Discrete ill-posed problems arise in connection with the numerical treatment of inverse problems, where one typically wants to compute information about interior properties using exterior measurements. Examples of inverse problems are image restoration and tomography, where one needs to improve blurred images or reconstruct pictures from raw data. This book describes new and existing numerical methods for the analysis and solution of rank-deficient and discrete ill-posed problems. The emphasis is on insight into the stabilizing properties of the algorithms and the efficiency and reliability of the computations., Presents an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, tomography, seismology, astronomy, and other areas., Here is an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, tomography, seismology, astronomy, and other areas. Rank-deficient problems involve matrices that are either exactly or nearly rank deficient. Such problems often arise in connection with noise suppression and other problems where the goal is to suppress unwanted disturbances of the given measurements. Discrete ill-posed problems arise in connection with the numerical treatment of inverse problems, where one typically wants to compute information about some interior properties using exterior measurements. Examples of inverse problems are image restoration and tomography, where one needs to improve blurred images or reconstruct pictures from raw data. This book describes, in a common framework, new and existing numerical methods for the analysis and solution of rank-deficient and discrete ill-posed problems. The emphasis is on insight into the stabilizing properties of the algorithms and on the efficiency and reliability of the computations. The setting is that of numerical linear algebra rather than abstract functional analysis, and the theoretical development is complemented with numerical examples and figures that illustrate the features of the various algorithms.

LC Classification NumberQA218 .H38 1998